Rules of Inference | Engineering Mathematics - Civil Engineering (CE) PDF Download

What are Rules of Inference for?

• Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements.
• An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “∴”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises.
• Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.

Table of Rules of Inference

Addition

• If P is a premise, we can use Addition rule to derive P∨Q.
  
  

Example

• Let P be the proposition, “He studies very hard” is true
• Therefore − "Either he studies very hard Or he is a very bad student." Here Q is the proposition “he is a very bad student”.

Conjunction

• If P and Q are two premises, we can use Conjunction rule to derive P∧Q.
  

Example

• Let P − “He studies very hard”
• Let Q − “He is the best boy in the class”
• Therefore − "He studies very hard and he is the best boy in the class"

Simplification

• If P∧Q is a premise, we can use Simplification rule to derive P.
  P∧Q/∴P

Example

• "He studies very hard and he is the best boy in the class", P∧Q
• Therefore − "He studies very hard"

Modus Ponens

• If P and P→Q are two premises, we can use Modus Ponens to derive Q.
  

Example

• "If you have a password, then you can log on to facebook", P→Q
• "You have a password", P
• Therefore − "You can log on to facebook"

Modus Tollens

• If P→Q and ¬Q are two premises, we can use Modus Tollens to derive ¬P.
  

Example

• "If you have a password, then you can log on to facebook", P→Q
• "You cannot log on to facebook", ¬Q
• Therefore − "You do not have a password "

Disjunctive Syllogism

• If ¬P and P∨Q are two premises, we can use Disjunctive Syllogism to derive Q.
  
  

Example

• "The ice cream is not vanilla flavored", ¬P
• "The ice cream is either vanilla flavored or chocolate flavored", P∨Q
• Therefore − "The ice cream is chocolate flavored”

Hypothetical Syllogism

• If P→Q and Q→R are two premises, we can use Hypothetical Syllogism to derive P→R
  

Example

• "If it rains, I shall not go to school”, P→Q
• "If I don't go to school, I won't need to do homework", Q→R
• Therefore − "If it rains, I won't need to do homework"

Constructive Dilemma

• If (P → Q) ∧ (R → S) and P ∨ R are two premises, we can use constructive dilemma to derive Q∨S.
  
  

Example

• “If it rains, I will take a leave”, (P → Q)
• “If it is hot outside, I will go for a shower”, (R → S)
• “Either it will rain or it is hot outside”, P ∨ R
• Therefore − "I will take a leave or I will go for a shower"

Destructive Dilemma

• If (P → Q) ∧ (R → S) and ¬ Q ∨ ¬ S are two premises, we can use destructive dilemma to derive ¬P∨¬R.
  
  

Example

• “If it rains, I will take a leave”, (P → Q)
• “If it is hot outside, I will go for a shower”, (R → S)
• “Either I will not take a leave or I will not go for a shower”, ¬Q ∨ ¬S
• Therefore − "Either it does not rain or it is not hot outside"